Development of an experiment for trapping, cooling, and spectroscopy of molecular hydrogen ions

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Development of an Experiment for

Trapping, Cooling, and Spectroscopy of

Molecular Hydrogen Ions

Dissertation

zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.), eingereicht am Fachbereich Physik der Universit¨at Konstanz

Universität Konstanz

von

Harald Schnitzler

Gutachter:

Prof. Dr. J. Mlynek Prof. S. Schiller, Ph.D.

Datum der m¨undlichen Pr¨ufung: 24. August 2001

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List of Figures

1.1 Appetizer . . . 5

2.1 Cryogenic Magnetic Trap . . . 12

2.2 Mechanical Analog to the Paul trap . . . 13

2.3 Typical Hyperbolic Paul trap . . . 13

2.4 Linear Paul trap . . . 14

2.5 Quadrupole Field in a Linear Paul Trap . . . 15

2.6 Stability Diagram of a Linear Paul trap . . . 17

2.7 Electrostatic Potential of the Endcap Electrodes . . 21

2.8 Field of the Endcaps in the Center of the Trap . . . 22

2.9 UHV Set-up CAD Design . . . 24

2.10 CAD Drawing of the UHV Chamber . . . 25

2.11 Experimental Set-up of the Linear Paul Trap . . . . 28

2.12 First Stability Regions and Pseudopotential Trap Depth . . . 29

2.13 Technical Construction of the Trap . . . 30

2.14 RF Trap Electronics . . . 31

2.15 Simulation of the RF Trap . . . 32

2.16 Cross-section of the UHV Set-up . . . 34

2.17 Electron Gun . . . 34

2.18 Set-up of the Trap and the UHV Chamber . . . 35

2.19 Loading Rate of the Trap . . . 37

2.20 Imaging with the CCD Camera . . . 38

2.21 Quantum Efficiency of the CCD Camera . . . 38

3.1 Graphical Picture of Doppler Laser Cooling . . . . 46

3.2 Light Pressure Force of a Single Beam . . . 48

3.3 Light Pressure of Counter-Propagating Beams . . . 49

3.4 Level Scheme of Be⁺ . . . 52

3.5 Relative Oscillator Strength . . . 53

3.6 Simulation of Sympathetic Cooling . . . 56

4.1 Sum-Frequency Generation in χ⁽²⁾ Media . . . 63

4.2 Phase Matching Curve of Type sinc²(∆kL/2) . . . 66

4.3 Index Ellipsoid of an Uniaxial Crystal . . . 69

4.4 Normal Surface the Biaxial Crystal LBO . . . 74

4.5 The LBO Crystal . . . 75

4.6 Set-up of the MOPA . . . 77

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4.7 Schematic of a Bow-Tie Cavity . . . 80

4.8 Sizes of the Beam Waists of the Cavity Mode . . . 83

4.9 Impedance Matching . . . 86

4.10 SFG Power versus Mirror Transmission . . . 89

4.11 Pound, Drever, Hall Phase and Frequency Stabilization 90 4.12 Set-up of the Lasers and Connected Feedback Circuits 92 4.13 Transfer Function of the Piezo . . . 93

4.14 Pictures of the Bow-Tie Cavity . . . 94

4.15 Stable Long-Term Operation of the SFG . . . 96

4.16 SFG Power for Variable Input Powers . . . 97

4.17 Frequency Tunability of the UV Light . . . 98

4.18 Frequency Drifts . . . 99

4.19 An Electro-Optical Phase Modulator . . . 101

4.20 Energy in the First Sidebands . . . 102

4.21 Technical Drawing of the EOM . . . 102

4.22 Measuring the Modulation Index . . . 103

5.1 Coordinates of the Molecular Hydrogen Ion . . . 107

5.2 Coupling of the Angular Momenta . . . 108

5.3 Born-Oppenheimer Potentials . . . 109

5.4 Two-Photon Spectroscopy . . . 116

5.5 Thermal Density Distribution at 300 K in HD⁺ . . 120

5.6 Population Transfer Using the STIRAP Method . . 121

5.7 Cross-sections of Selective Photodissociation of HD⁺ 123 5.8 Measurement Cycle . . . 124

5.9 Experimental Set-up for Two-Photon Spectroscopy 125 5.10 Absolute Frequency Measurement (A) . . . 126

5.11 Absolute Frequency Measurement (B) . . . 126

5.12 Optical Fluorescence Detection of HD⁺ . . . 129

6.1 Overview of the Apparatus . . . 131

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List of Tables

2.1 Sizes and Parameters of the Linear RF Trap . . . . 28

2.2 Data of the Beryllium Source . . . 35

3.1 Parameters of the Plots . . . 48

3.2 Data of the Beryllium Ion . . . 51

4.1 Characteristics of LBO . . . 73

4.2 ABCD Matrices . . . 81

4.3 Specifications of the Cavity . . . 83

4.4 Internal Losses in the Cavity . . . 86

4.5 Characteristics of the EOM . . . 103

5.1 Relevant Energy Levels in HD⁺ . . . 113

5.2 Ro-Vibrational Transition Moments . . . 115

5.3 Ro-Vibrational Lifetimes in HD⁺ . . . 115

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Chapter 1 Introduction

In the past few years trapping and cooling of particles has developed into one of the most exciting fields of modern physics. The fundamental work on trapping of atomic particles and on manip- ulating the motion of atoms, especially with light forces, has been honored with two Nobel-Prices: 1989 for Dehmelt and Paul, as as well as 1997 for Chu, Phillips and Cohen-Tannoudji.

However, the past work was mainly restricted to atoms. The experiment described in this thesis transfers the accumulated knowledge on cooling of atoms to molecules. The combination of appropri-

ate experimental methods allows one to trap and to cool molecules Trapping, cooling, and spectroscopy of molecules and to finally perform spectroscopy of cold molecules with unprece-

dented precision. This opens up a new field of research, which should be of interest for physics and physical chemistry and will make possible new applications in biology, medicine, and environ- mental research.

1.1 Quantum Metrology and Funda- mental Tests of Physics

The molecular hydrogen ion as the simplest molecule at all – con- The hydrogen molecular ion sisting of two nuclei and one electron facilitating the molecular bind-

ing – is outstandingly suitable for theoretical calculations which can be compared to experimental data. In addition to H⁺₂, its isotopomers HD⁺ and D⁺₂ have played important roles in the development of molecular quantum mechanics. They are used for the description of many different molecular theories, methods, and ap- proximations. The fact that only one electron exists and thus no interaction between electrons needs to be taken into account, allows the observation of other aspects in molecular structures. Thus the structure and dynamics of the molecular hydrogen atom are probably better understood than those of all other molecules.

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The highly accurate spectroscopic measurement of the rotational vibrational transitions in the molecular hydrogen ion can be used for

• the metrology of fundamental constants,

• tests of time invariance of fundamental constants,

• validation of molecular theories,

• to establish an molecular frequency standard based on one of the simplest molecules.

The measurement of fundamental constants is an important branch Metrology

in high precisionmetrology and of outstanding physical interest. On the one hand it is used for the definition of standards, on the other hand the techniques developed here are often transferred to other fields of physics.

The values of fundamental constants given in this thesis are taken from the 1998 adjustment of the fundamental constants [1]. Each value and the corresponding uncertainty is deduced by consideration of several experiments and can thus not be related to a single measurement.

In the past, laser-spectroscopy has already been used successfully for precision measurements of fundamental constants. The Rydberg constantR∞ is actually the best known fundamental constant with a relative uncertainty of 7.7 · 10⁻¹². It is deduced from several experiments on the metrology of the hydrogen atom [2].

One set of fundamental constants are the masses of fundamental particles. Generally, it is only possible to measure ratios between two masses; the ratio between the mass of an electron and the proton,me/mp, is the most fundamental and is known to a relative uncertainty of 2.1· 10⁻⁹. The most accurate value of this ratio has been measured using the cyclotron resonance frequencies in a Penning-trap [3].

One goal of our project is the first measurement ofme/mp based on laser spectroscopy of vibrational transitions in HD⁺. Once theory and experiment both achieve an accuracy better than 2.1·10⁻⁹, the spectroscopic values of the ro-vibrational transition frequencies of HD⁺can be used to obtain a more accurate value of the fundamental constant m_e/m_p.

An absolute measurement of the me/mp ratio by measuring the frequency of the molecular transition is not trivial. It has to be done by comparison against other optical frequency standards. The frequency difference between two vibrational levels is given by

∆νvib = R∞

c

rme

mp ·f(me

mp

, α, . . .). (1.1)

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1.1 Quantum Metrology and Fundamental Tests of Physics 3

This expression is mainly the Born-Oppenheimer-approximation in higher order, where the function f is only weakly dependent on other quantum numbers. Further improvement of theoretical calculations is required to calculate f to the desired accuracy. Due to the – in terms of molecular theory – relative simplicity of the molecular hydrogen ion, highly precise (and complicated) calculations exist, which shall be significantly improved in cooperation with theorists in Moscow, Sofia and Southhampton. Using the Ry- dberg constant given above, one can extract a new value for the mass ratio of me/mp from a measured value of ∆νvib with a so far unachieved relative accuracy in the region of 10⁻¹⁰.

In a second step a more accurate value ofme/mp can also be used to find a more accurate value of the fine-structure constant α, which is given by

R∞ = mec h

α²

2 → α =

s 2R∞

c h mp

mp

me

. (1.2)

At the momentαis known to a relative accuracy of 3.7·10⁻⁹, while h/mp is known to 5·10⁻⁷. Experiments with atom interferometers in Stanford and a proposed satellite experiment (HYPER) which should improve the accuracy of h/mp significantly.

In addition, if a very high stability of the HD⁺ transition frequen- Test of time invariance cies can be achieved, precision measurements of physical constants

can also be used for fundamental tests of physics and for validity checks of physical theories, e. g. tests of time invariance of fundamental constants. A violation of the time invariance would contra- dict to the principle of equivalence of the general theory of relativity.

However, the theory of relativity is not complete because it lacks quantization. This justifies the tests of fundamental masses as a test of the general theory of relativity. Other theories (e. g. string theories) predict time dependencies of fundamental constants. By measuring upper limits of time variations, those theories can be validated or contradicted (more likely) and perhaps bring us a bit closer to a complete theory.

These kind of tests were already done: The fine structure constant α as an unitless constant provides an ideal physical quantity for a constant time variation test, independently of any possible units time variations. The currently best laboratory determination of the upper bound of the relative time dependence ofα consists in a long time (typically several months) comparison of two frequency standards: an electronic transition and the resonance frequency of an optical cavity, which depend on the fundamental constants in a algebraic different way. The electronic transition frequency depends

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on the Rydberg energy

νelectronic ∝ R∞

c ∝mecα², (1.3)

while the resonator frequency depends on νresonator ∝ h

length ∝ h

a0 ∝mecα . (1.4) The ratio of these two frequencies is proportional to α, and sets an upper limit to the relative time dependence of this constant.

The most accurate test so far was done by comparing a microwave- frequency of a hyperfine transition of atomic ions in a Paul trap with the hydrogen maser. The found upper limit for the time dependency of the fine-structure constant is |dα/dt|/α <4·10⁻¹⁴/year [4]. For further improvements on this, new oscillators with higher frequency stability need to be developed.

The actual upper limit of a possibly time variance of m_e/m_p is one order of magnitude lower than the one of α. For the experimental setup of this test an ultra-stable oscillator can be implemented, which is stabilized on a vibrational transition (e. g. of HD⁺). For comparison another oscillator has to be implemented which is based on an electronic transition. The ratio of the frequencies depends onm_e/m_p only. After consideration of systematic effects and measuring with sufficiently high accuracy, this would yield an improved upper limit on the time variance of the electron-proton mass ratio.

Alternatively, a macroscopic reference oscillator can be applied for the frequency comparison. To this end, we can rely on the knowledge on cryogenic oscillators in our group, which have about the best frequency stability in the world (in the range between 10- 1000 s). In this case the time variance of the productα·p

m_e/m_p is tested. However, in combination with the upper limit of|dα/dt|/α given above, this can be used to improve the accuracy of ^d(m_m^e^/m^p^)/dt

e/mp . More generally, very high precision measurements on molecules can Validation of

molecular calculations

provide improved tests of molecular theories like molecular quantum electrodynamics (QED) and relativistic corrections. The comparison between spectroscopic measurements and theoretical predictions as a test of molecular theories and calculation methods is of special interest. The targeted accuracies will allow a first test of radiative (QED) and relativistic corrections in molecules. Due to the complexity of the calculations, such kind of tests are preferably performed on the simplest molecule!

If stable long term operation of the hydrogen spectroscopy is Molecular

frequency standard

achieved, this is of fundamental interest for the implementation of a frequency standard based on the simplest molecule.

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1.2 Overview of the Project 5

There is also great astrophysical interest in the exact knowledge of the spectral lines of the molecular hydrogen ion. New models in astrophysics predict molecular hydrogen ions in interstellar clouds, but these could not yet be verified. The detection would be sim- plified if highly precise values of the transition frequencies of low vibrational levels were known. Ultimately, the precise knowledge of the density of molecular hydrogen ions in interstellar clouds would allow a verification of existing models for formation of interstellar molecules.

1.2 Overview of the Project

The homonuclear species H⁺₂ and D⁺₂ do not possess electric dipole moments, therefore they cannot exhibit electric-dipole allowed vibrational or rotational spectra. Only the mixed isotope species,

HD⁺, in which the symmetry is broken, exhibits an infrared spec- Why HD⁺ trum of rotational vibrational dipole allowed transitions. Conse-

quently, HD⁺ has been selected for the experiment.

The theoretical treatment of the molecular hydrogen ion is (to a certain accuracy) much easier than its experimental investigation.

Despite its high binding energy and its thermodynamic stability, it is very reactive. Experimentally, the particles need to be separated e. g. from neutral hydrogen. Studies of electronic transitions are difficult: the first exited electronic state is mainly repulsive and higher exited states are more than 11 eV above the ground level, making them hardly accessible for spectroscopy.

Fig. 1.1 schematically shows the experiment.

e - Be

molecules

cooling spectroscopy photodissociation

CCD camera

ion detector

laser beams for

Figure 1.1: “Appetizer”. Schematic view of the experiment. HD⁺ molecules are confined in a linear rf Paul trap and cooled sympathetically by laser cooled Be⁺. High-resolution 2-photon spectroscopy of HD⁺ is proposed, which can be detected by selective photodissociation, using the ion detector.

The HD⁺ molecular ions are confined in a linear rf Paul trap. Trapping and cooling of HD⁺

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Compared to a Penning trap the linear rf Paul trap has the main ad- Linear Paul trap

vantage of not requiring any magnetic field, since such a field could be a problem for high resolution spectroscopy. Instead of a classi- cal hyperbolic shaped trap, we will use a linear trap design. This configuration provides more trapped molecular ions in the free-field region, and thus minimizes the influence of second order Doppler shifts.

While trapping of molecular ions is straightforward, the rf motion of the ions causes them to heat up to a temperature as high as several eV, equivalent to more than 10000 K. The corresponding Doppler broadening (first and second order) precludes precise spectroscopic measurements.

It is therefore necessary to cool the particles to mK-temperatures, which we intend to achieve by sympathetic cooling. Another reason Sympathetic

cooling for cooling the molecules is to populate the vibrational ground state level v = 0.

Sympathetic cooling relies on the long-range Coulomb interaction between the molecular ions and the laser cooled atomic ions which are confined in the same trap.

The method of sympathetic cooling is very universal. It should allow to cool nearly any molecular ion: small molecules which are of fundamental interest for physicists as well as large molecules which might be of fundamental interest for biology or chemistry.

It is known that sympathetic cooling works best for minimum mass differences between the interacting particle species. Since HD⁺ has only a mass of 3 a.u., we have thus chosen Be⁺as the lightest atomic ion that can be laser cooled.

The light for laser cooling of Be⁺ at 313 nm is produced by doubly Doubly resonant

sum-frequency generation

resonant sum-frequency generation in LBO. The light from a frequency doubled Nd:YAG laser at 532 nm and from a diode laser at 760 nm is resonated in a single bow-tie cavity. The cavity is locked to the Nd:YAG laser, while the diode laser in turn is locked to the cavity. This method transfers the intrinsic high frequency stability of the Nd:YAG laser to the generated UV-light. Output powers >2 mW are obtained with 1.4 W at 532 nm and 11.5 mW at 760 nm. The system also allows a wide continuous tunability of 16 GHz in the UV, which is required for fast and efficient cooling of the initially hot ions.

Under appropriate conditions the ions form a new state of matter Ion crystal

in the trap: at sufficient high cooling rates the ions stop moving around each other and stay in an ordered structure. It was demonstrated, that non laser cooled atomic ions can be added to such an ion crystal (sympathetic crystalization) [5, 6].

The goal of this project is to embed for the first time a significant Unique

environment for the molecules

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1.2 Overview of the Project 7

number of molecular ions in a laser cooled atomic ion crystal and to cool them sympathetically. These molecules exist in an unique environment. They

• have very low kinetic energies (mK) which is transferred to their inner degrees of freedom,

• are isolated from perturbations by electric or magnetic fields:

by the use of a linear radio frequency trap, we can avoid an inherent magnetic field and due to the geometry of the trap the electric field on the axes of symmetry vanishes,

• are free of collisions with walls and interactions with the residual gas can be neglected by working in an ultra-high vacuum of 10⁻¹⁰ mbar or better,

• the distance to the next neighbors in an ion crystal is about 10 µm,

• the lifetime of the trap is in the region of a couple of minutes due to the high trapping potential of a few eV,

• the ions may be held in interaction with the laser beam for a long time, minimizing transit-time broadening.

• For these reasons, spectroscopy on the molecules can be performed with so far unprecedented spectral resolution (>

1:10¹⁰) which will allow for new fundamental tests of physics and the examination of new or unknown static and dynamic properties of molecules.

In combination with suitable laser sources an ultra-cold ensemble Spectroscopy of cold molecules of molecules can lead to spectroscopic measurements with unprece-

dented spectral resolution.

To avoid the small but, despite of the cooling, still significant 1st

order Doppler-broadening, we propose 2-photon spectroscopy of the 2-photon spectroscopy (v = 0, N = 4)−(v = 4, N = 4) rotational vibrational transition

of HD⁺. In comparison to saturated absorption spectroscopy, the advantage is that all the molecules contribute to the signal. The light needed for the 2-photon transition spectroscopy at 2.8 µm will be produced by an optical parametric oscillator (OPO), which is developed in other projects of our group.

A difficulty of the spectroscopy is due to the low excitation rate of approximately one per second per molecule, which is not enough

to detect a fluorescence signal. In order to get a signal, selective Selective photodissociation photodissociation will be used. Owing the Franck Condon factor,

the cross section depends on the wavelength of the light used for the dissociation and on the vibrational level in which the molecules originate.

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1.3 State of Research

1.3.1 State of spectroscopic investigations

Several spectroscopic investigations of the the molecular hydrogen Spectroscopy of

the molecular hydrogen ion

ion can be found in the literature.

Already in the 60ies the group of Dehmelt demonstrated alignment . . . in Paul traps of the H⁺₂ molecular ion in a Paul trap by selective photodissociation

and spectroscopy of its radio-frequency spectrum [7].

Wing et al. reported the observation of the infrared spectrum of . . . in ion beams

HD⁺ in an ion beam in 1976. Several rovibrational transitions of the lowest vibrational energy levels were measured [8].

Complementary, in the 80ies Carrington and Buttenshaw performed vibration-rotation spectroscopy of the HD⁺ ion near the dissociation limit [9, 10]. They used selective photodissociation in the infrared and mass selective detection of the fragments, to detect the vibrational transitions [11]. Hereby they were also able to determine the relative cross sections of photodissociation of H⁺₂, D⁺₂ and HD⁺ [12].

Very recently, in February 2001, a direct measurement of a pure rotational transition in the v = 19 level of H⁺₂ was reported at 14961.7±1.1 MHz, using a modified version of a fast ion beam spectrometer [13]. Recent theory predicts significant electric dipole intensity in forbidden rotation and rotation-vibration transitions involving levels near the dissociation limit. The measurement is consistent with this value.

1.3.2 Theoretical Predictions

The hydrogen molecular ion is the simplest molecule [14], its electronic Hamiltonian is exactly soluble within the Born-Oppenheimer approximation, and its vibration-rotation energies are the most ac- curately calculated ones of any molecule.

The non-relativistic energies of the lower vibration-/rotation-levels Non-relativistic

calculations are calculated with variational-algorithms to relative uncertainties of 10⁻¹⁵. For levels with higher energy the accuracy decreases only slightly [15].

However, already for low accuracies relativistic and QED- Relativistic and

QED corrections corrections have to be taken into account carefully. Relativistic corrections of order α² of the Breit-equation were calculated [16], and lead to corrections of order of 10⁻⁵, with a uncertainty of 10⁻⁸; more accurate calculations are possible. Vertex-corrections in the orderα³ lead to corrections in the order of 10⁻⁷ and could be calculated to 10⁻¹². To estimate corrections in the order of α⁴ methods

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1.3 State of Research 9

are currently being developed [17]. The physics of higher corrections is not yet worked out. Herein the theory will be challenged.

In the collaborations mentioned above, progress in the calculations of the hydrogen molecular ion will be pursued.

Today, theory and experiment are in agreement to an accuracy of Agreement between theory and experiment 10⁻⁶ for the hydrogen molecular ion. However, the last experimen-

tal data derive from the year 1976 [8] and are several orders of magnitude less accurate than actual theoretical calculations. Espe- cially the hyperfine structure and QED-effects of the low vibrational levels could not be observed yet.

1.3.3 Trapping and Cooling of Molecules

The particles in the experiments on ion beams have kinetic en- 1st and 2nd order Doppler effects ergies of several thousand eV and a thermal energy distribution.

The scanning of the transition frequencies is done by using a spectroscopy laser collinear with the ion beam and shifting the particles resonance frequency via the Doppler-shift, i. e. by changing their mean kinetic energy. Thus, the spectroscopic results are Doppler- broadened and -shifted, and the relative accuracy is only less better then 10⁻⁶. Even if the first order Doppler-shift is eliminated, the short interaction time of the ions with the laser beam would lead to transit time broadening of the same order.

In our project we avoid these types of broadening by trapping and cooling the particles. Furthermore, we plan to eliminate the first- order Doppler-broadening completely by performing 2-photon spectroscopy. The remaining second-order Doppler-broadening is a relativistic effect and independent of the direction of movement: it is proportional to the kinetic energy of the ions and will be seriously reduced by sympathetic cooling of the molecular ions in the trap.

Trapping of cold molecules in a magnetic trap has been demon- Magnetic trap strated lately by Doyle et al. at Harvard [18] (→2.1). In their ex-

periment laser ablated paramagnetic molecules are cooled by cold helium buffer gas to a temperature below 1 K and then trapped.

This kind of trapping can only be applied on the certain fraction of magnetic molecules. Moreover, the magnetic trap intrinsically comes along with a strong magnetic field, which, for our goal, might harm the applicability for comparison to theory and thus for fundamental tests of the molecular physics.

In our project we use a different approach: molecular ions are stored Our trap by an electric alternating current in a rf Paul trap, first demon-

strated at [19]. This leads to a stored plasma of ions with a temperature in the region of ∼ 10000 K. The ion plasma of atomic and molecular ions can be cooled by laser cooling of the atoms and

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thereby sympathetically cooling the molecules to a temperature in the mK region.

Laser cooling of trapped ions was first proposed 1975 by Wineland Laser cooling

and Dehmelt [20] and experimentally demonstrated in a Paul trap in 1978 [21]. Temperatures in the sub-mK range have been observed for direct laser cooled atomic ions in Paul traps. A single atomic ion has even be cooled down to the ground state of motion of the trap potential [22].

The method of sympathetic cooling of atomic ions was first demon- Sympathetic

cooling strated by Larsonet al. [23]; Baba and Waki shortly reported sympathetic cooling of molecular ions which were ionized from the residual gas and cooled to about 10 K [24].

1.4 In this Work

The fundamentals of ion trapping in a linear rf Paul trap are described in chapter 2, presenting the experimental design and set-up of the trap, the X-UHV environment and the connected peripherals.

Chapter 3 reviews various possibilities of cooling; laser cooling of ions in a Paul trap, in particular laser cooling of Be⁺, and sympathetic cooling of molecules is considered in further detail.

In chapter 4 we present a novel approach for the generation of highly frequency-stable, widely tunable, single-frequency cw UV light suitable for laser cooling of Be⁺. Sum-frequency generation (SFG) of two solid-state sources using a single cavity resonant for both fundamental waves is employed. Using a highly stable, narrow linewidth frequency-doubled cw Nd:YAG-laser as a master laser and slaving to it the SFG cavity and the other fundamental wave from a diode laser, we generate UV radiation of > 2 mW output power around 313 nm. We obtain a coarse tuning range of 6 nm, a continuous tunability of 16 GHz, a sub-MHz linewidth, a frequency drift below 20 MHz/h, and stable long-term operation. The theory of doubly-resonant SFG is also given.

A detailed concept of ultra-high precision spectroscopy of the HD⁺ molecule is worked out in chapter 5 and discussed quantitatively.

The fundamentals of the molecular theory as well as the spectroscopic techniques, 2-photon spectroscopy in particular, are explained.

The experimental and conceptual work of this thesis is summarized in chapter 6. A final conclusion and a discussion of the future experimental steps is given.

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Chapter 2 Trapping Particles

High resolution spectroscopy of atoms and molecules requires a cold confined ensemble of the particles of interest. This allows long interaction times and minimizes the first and second order Doppler shift.

In this chapter a short general overview of different trapping types is given at first. This should motivate our specific choice of trap, the linear radio frequency (rf) Paul trap , which is discussed in (→2.2).

The last section of this chapter describes the complete experimental set-up of the trap in an ultra-high-vacuum (UHV) environment and the associated electronics.

2.1 Different Types of Traps

A large variety of neutral atoms can be trapped in magneto-optical Magneto-optical and optical dipole traps traps and optical dipole traps. Unfortunately, those traps are lim-

ited to particles with a simple effective two-level electronic structure. In molecules these mechanisms for laser trapping and laser cooling are not applicable, due to the ro-vibrational energy level structure. The lack of a general cooling method has so far pre- vented trapping of most particles.

Lately cold paramagnetic particles were trapped magnetically by Doyleet al.at Harvard [18]. This type of trap (Fig. 2.1) is also being set up in a newly started project in our group at the university of Konstanz [25].

Laser ablated molecules are collisionally cooled by a buffer gas of Magnetic quadrupole trap cryogenic helium to a temperature below 1 K. They are captured

in a strong magnetic quadrupole trap in a ³He/⁴He dilution refrigerator. The phase space density of the trapped sample can then be increased by evaporative cooling. This kind of trapping can be applied to neutral paramagnetic molecules and atoms. Proposed applications of this trap include tests of fundamental physics through

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1 2 3 4 5 6 Tesla

234 mm

Figure 2.1: Cryogenic magnetic trap for ultra cold atoms and molecules.

precision spectroscopy on ultracold atomic and molecular samples as well as the production of degenerate Bose and Fermi gases.

However, the spectroscopic applicability for fundamental tests is af- fected by the strong intrinsic magnetic field of the trap. The influence of the magnetic field leads to large uncertainties in theoretical calculations of molecular energy levels. Additionally, the set-up of a dilution refrigerator with a superconducting magnet represents a major experimental effort.

Charged particles in general can be trapped by electric and mag- Ion traps

netic fields. Unfortunately, the electrostatic the Laplace-equation

∇²Φ = 0 (2.1)

shows that there is no electrostatic field which has a potential minimum in all three dimensions of space: assume the harmonic potential

Φ∝¡

αx²+βy²+γz²¢

, (2.2)

then the Laplace-equation (2.1) yields α+β+γ = 0. α, β, and γ cannot have the same sign simultaneously. Consequently, any electrostatic field (6≡0) is always attractive in at least one direction and thus cannot give electrostatic confinement in all three dimensions.

Nevertheless, in principle there are at least two ways of trapping Penning trap

charged particles. One is found by overlapping of electrostatic and magnetostatic fields, which leads to Penning traps [26, 27]. These traps can confine any kind of ions. Unfortunately, with respect to the pursued fundamental tests of physics, they bring in an intrinsic magnetic field which we like to avoid, as mentioned above.

In this thesis we will focus on the other possibility to trap ions, Paul trap

which is given by the so called Paul trap [19, 26, 27, 28]. These traps use an alternating electrical potential between two electrodes to confine the ions. To explain this dynamic trap potential Paul in- troduced a mechanical analog: a mechanical realization of a static

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2.1 Different Types of Traps 13

saddle potential, as shown in Fig. 2.2, fulfills Laplaces’ law. Con- finement of an iron ball can be achieved by rotating the saddle potential with a suitable angular frequency which depends on the particles’ mass.

Figure 2.2: Mechanical analog to the Paul trap: if the saddle potential is rotated with a suitable frequency, it can confine the particle in its center.

Technically, this is implemented by applying a radio frequency voltage

Φ0 =U −V cos Ωt (2.3)

to their hyperbolically shaped electrodes, as shown in Fig. 2.3 (“rf trap”).

F0

Figure 2.3: Typical hyperbolic Paul trap.

For usual electrode sizes with 2z²₀ = r₀² the electrostatic potential is described by a quadrupole field with α = β = 1, γ = −2 (2.2), which in cylindric coordinates is given by

Φ = Φ₀r²−2z²

r₀²+ 2z₀² (withr² =x²+y²). (2.4) Depending of the sign of the charge this gives confinement either in thexy-plane or in z-direction only.

If one applies a rf voltage as in (2.3), the attracting potential starts cycling between the two electrodes and one could expect that the

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time averaged potential vanishes. This is true for a homogeneous field distribution. However, in the time averaged inhomogeneous quadrupole field a small mean force remains, which is directed to- wards the field strength minimum i. e. to the center of the trap.

The equations of motion can be solved analytically [28]. They lead to stability diagrams, which show regions of stable trajectories of the particles depending on their charge/mass ratio and the applied voltages and the size of the trap.

Compared to the types of traps mentioned before, due to the ab- sence of magnetic fields and strong coupling of ions by the Coulomb force to external electric fields, the Paul trap minimizes disturbances of the inner structure of the trapped particles. In our experiment we use a linear Paul trap, which will be described in detail in the next section. It is slightly different from the hyperbolic set-up just described, which brings in further improvements for applications dealing with ultra high resolution spectroscopy.

2.2 Trapping Ions in a Linear Paul Trap

The linear Paul trap [29] uses a combination of radio frequency and static electric potentials to create a potential which can trap any type of charged particles, including atomic ions as well as molecular ions and also macroscopic particles. The arrangement of the electrodes is shown in Fig. 2.4.

Fec, 0

2r’

2r’ 2z0

~

^F^F⁰⁰

z x

y

2r0

Figure 2.4: Linear Paul trap.

In comparison to the hyperbolic Paul trap described above, the linear Paul trap confines more ions at less mean kinetic energy.

That is explained by the linear symmetry: while in the hyperbolic set-up there is just a single point in the trap center where the rf

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2.2 Trapping Ions in a Linear Paul Trap 15

field vanishes for all t, for the linear set-up this is the case along its whole symmetry axis (see rf heating (→2.2.4)). With respect to spectroscopy, this leads to less Doppler broadening and thus to better accuracy of spectroscopy.

In the following the physics of the rf trap are described in more detail.

2.2.1 Equation of Motion in a Two-Dimensional Quadrupole

Confinement in thexy-plane of the trap is produced by the rf voltage, which is applied, in opposite phases, to the two orthogonal pairs of the trap electrodes (Fig. 2.5).

F / 2₀ F / 2₀

- F / 2₀

- F / 2₀ 2r0

x y

z

Figure 2.5: Quadrupole field in the xy-plane of a linear Paul trap.

The resulting electric field is a two-dimensional quadrupole field, as given in (2.2), with α= 1, β =−1, γ = 0:

Φ = Φ0

x²−y²

2r₀² . (2.5)

Now we consider an ion of mass m and charge Q, in an rf voltage as in (2.3) is applied to the electrodes. The equations of motion in the field (2.5) are given by

¨ x+ Q

mr²₀(U −V cos Ωt)x= 0, (2.6)

¨ y− Q

mr₀²(U−V cos Ωt)y= 0.

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With the definition of some handy dimensionless parameters τ = Ωt

2 , a=ax =−ay = 4QU

mr²₀Ω² , q=qx =−qy = 2QV

mr₀²Ω² (2.7) one rewrites the equations of motion (2.6) as

∂_τ²u+ (au−2qucos 2τ)u= 0, u=x, y , (2.8) which can be identified as the generalized form of the Mathieu differential equation.

Its general solution can be written as u(τ) =α⁰e^µτ

n=∞X

n=−∞

C_2ne^2inτ +α⁰⁰e^−µτ

n=∞X

n=−∞

C_2ne^−2inτ, (2.9) where the constants α⁰ and α⁰⁰ depend on the starting conditions u0, ˙u0 and τ0, while C2n as well as the characteristic coefficient µ depend on the parameters a and q only.

The characteristic coefficient µ determines whether the Mathieu equation gives stable or unstable solutions. A solution is called stable if the amplitude ofu – which describes the particle’s trajectory – stays finite for τ → ∞. µ is only depended on the parameters a and q, not on the initial values.

• If Re(µ)6= 0, one of the factors e^±µτ diverges and the trajectory is unstable.

• Suppose µ is purely imaginary: µ = iβ. If β is not an integer, the results are periodic, stable trajectories. In- teger βs set boundaries between stable and diverging solutions. In the Mathematica 4 software the characteristic values a for these even(odd) Mathieu functions of nth order and parameter q can be obtained by the functions a=MathieuCharacteristicA(B)[n, q]. The resulting regions of stable parameter pairs (a, q) are shown in Fig. 2.6.

Only those pairs (a, q), for which stable trajectories in x and y dimension are found, lead to stable trapping. Experimentally the region with lowest voltages is the easiest to access. Thisfirst stability regionis shown on the right-hand side of Fig. 2.6. The diagram shows that stable trapping without rf voltage (i. e. q = V = 0) is not possible. On the other hand, if a pure rf voltage is applied (i.

e. a=U = 0) stability is given for all ions with

q < qmax= 0.908 (2.10)

⇔ m > mmin = 2QV 0.908r₀²Ω² .

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2 4 6 8 10 q

-4 -2 2 a 4

xstable

ystable

0.2 0.4 0.6

0.1

0 0.2

0.8 a

q q1 q2

qmax

Figure 2.6: Stability diagram of the ions in a linear Paul trap. Values (a, q) in the gray shaded areas lead to stable trajectories in either x or y direction. For stable trapping both axes need to be stable. An enlargement of the first stability region is shown on the right-hand side.

In this case the trap works as a high-pass mass filter.

Fixed values of the voltages U and V lead to a mass independent Quadrupole mass spectrometry value of a/q = 2U/V. For a given ratio U/V all masses hence lie

on a straight working line in the (a, q)-diagram, as drawn in the right-hand side of Fig. 2.6. Only those masses withq1 < q < q2 are trapped. By scanning U and V with fixed ratio U/V one brings in one mass after another into the stability region. This operation of the two-dimensional quadrupole is known as quadrupole mass spectrometry.

2.2.2 The Pseudopotential

In the previous section a criterion for the stability of the particles’ trajectories was deduced from the equations of motion in a 2-dimensional quadrupole field. In this section a time averaging of the quadrupole field is performed, which leads to a quantitative understanding of the trapping potential which is namedpseudopo- tential.

To find an expression for the pseudopotential an one-dimensional motion of the ion in the applied electric potential is considered [30, 31, 32]. The potential consists of two components: a static component UF(x) and an oscillating component Uf(x), which give rise to a static force F(x) and an oscillating force f(x, t), respec- tively,

F(x) =−QdUF(x)

dx (2.11)

f(x, t) =−QdUf(x, t)

dx =f0(x) cos(Ω(t−t0)). (2.12)

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where x=X(t).

On time scales longer than 1/Ω the particle will move in a smooth path which is the secular motion S(t). The force f(x, t) adds an oscillation at the frequency Ω to the path S(t) which is named micromotion M(S, t). The trajectory of the particle is thus given by

X(t) = S(t) +M(S, t). (2.13) Assume that the amplitude of the micromotion is small compared to the amplitude of the secular motion S À M, while the velocity of the micromotion is fast compared to the velocity of the secular motion ˙S ¿ M˙. M can then be considered as perturbation to S, and the equation of motion

mX¨ =F(X) +f(X, t) (2.14) can be expanded in M around the path described byS

m³

S¨+ ¨M´

=F(S) +f(S, t)+ (2.15)

M ∂x(F(x) +f(x, t))¯¯¯

x=S+ 1

2M²∂_x²(F(x) +f(x, t))¯¯¯

x=S+ O(M³).

Using only the first order in M and assuming that the large and rapidly oscillating terms on each side are dominating and therefore approximately equal, we obtain

mM¨ ≈f(S, t), (2.16)

which can be integrated by using f(x, t) from (2.12) M(S, t)≈ 1

mΩ²f0(S) cos (Ω (t−t0) +π) . (2.17) The micromotion M is oscillatory with the frequency of the force f, but with a phase difference of π, and an amplitude depending on the coordinate of the secular motion.

The mean kinetic energy of the micromotion becomes Energy of the

micromotion

Kmicro(x) = f₀²(S)

4mΩ² . (2.18)

An expression for the slow secular motionS(t) is found by averaging first order of (2.15) over one period T = 2π/Ω of the micromotion.

The fast oscillating terms containing M, ¨M, and f disappear, except for M ∂xf(x, t)¯¯¯

x=S which contains a cos²: mS¨=F(S) + 1

T Z T

0

M ∂xf(x, t)¯¯¯

x=S dt . (2.19)

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Using f and M from (2.12) and (2.17) this results in mS¨=F(S)− 1

2mΩ²f0(x)∂xf0(x)¯¯¯

x=S, (2.20)

which, expressed in the potentials defined in (2.11) and (2.12), can be written as

mS¨=−∂x

µ

QUF(x) + Q² 4mΩ²

¡∂xUf(x)¢2¶ ¯¯¯

x=S. (2.21) This equation shows that the particle – when time averaged over the fast oscillations – in addition to the assumed static potential UF feels another quasi-static potential raised byUf, which is the so

called pseudopotential Pseudopotential

Ψ(x) = Q² 4mΩ²

¡∂xUf(x)¢2

. (2.22)

Because of the quadratic dependence on both Q and the potential derivative, the pseudopotential will repel the particle from regions with high amplitudes of the oscillatory field, independent of the sign of the particles’ charge:

The ion in an oscillating field is a low field seeker. Low field seeker We note that the pseudopotential is also equal to the mean kinetic

energy in the micromotion given in (2.18).

With the definition of the pseudopotential the secular potential can Secular potential be written as

Ψsec(x) =QUF(x) + Ψ(x). (2.23) The derived potentials (2.22) and (2.23) can be easily generalized to three dimensions.

The time dependent electric potential of the two-dimensional quadrupole field in our trap (Fig. 2.5) is given by

Φ(x, y, t) = V

2r²₀ (x²−y²) cos(Ωt). (2.24)

With (2.22) the corresponding pseudopotential of the two- Resulting pseudopotential of the quadrupole field dimensional quadrupole field yields

Ψ(x, y) = Q²V²

4mΩ²r₀⁴(x²+y²). (2.25) A trapped ion will be lost if it reaches the electrodes, so the the pseudopotential depth is

Ψ0 = Ψ(r0) = Q²V²

4mΩ²r²₀ = QV

8 q . (2.26)

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The secular potential of the two-dimensional quadrupole field, Resulting secular

potential of the quadrupole field

which in addition to the pseudopotential takes also the electrostatic potential U into account, is given by

Ψsec(x, y) = Q²V²

4mΩ²r⁴₀ (x²+y²) + QU

2r²₀(x² −y²), (2.27)

with in xand y direction different secular potential depths of

Ψ_sec,0 = Q²V²

4mΩ²r²₀ ± QU

2 . (2.28)

Finally we find a secular oscillation frequency

ωsec =

s2Ψsec,0

mr₀² =

s Q²V²

2m²Ω²r₀⁴ ± QU mr²₀ = Ω

2 rq²

2 ±a . (2.29)

In this section the potential in the xy-plane of the trap was considered. To achieve confinement in all dimensions a trapping force in the z direction of the trap needs to be added which is discussed in the following section.

2.2.3 Longitudinal Confinement

Confinement of the ions in z direction can be achieved by electrostatic fields. To this end, the potential at both ends has to be raised by a certain amount compared to the potential in the center of the trap. Different electrode setups (pin electrodes on the axis, ring electrodes, etc.) have been demonstrated [33]; to provide best optical access – especially along the axis of the trap – we decided to use a set-up with galvanically separated endcap electrodes. There- fore each of the four electrode rods is divided into three electrically isolated segments. If an electrostatic potential of Φec,0 is applied to the eight endcap electrodes a static trapping potential as shown Fig. 2.7 is achieved.

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FFecec, 0(0,0,) /z 0,0 0,2 0,4 0,6 0,8 1,0

mm distance from trap center

-40 -30 -20 -10 0 10 20 30 40

Figure 2.7: Electrostatic potential along the z-axis of the linear Paul trap. The outer (endcap-) electrodes have a mean potentialΦec,0 while the center electrodes are at 0.

The electric field distribution was simulated with finite elements using theSimion 6software. We find that the longitudinal potential for our specific trap dimensions (which are discussed in section 2.3) has steep walls (compared to the trap length). While the potential on the axis in the endcap region reaches approximately Φec,0it drops rapidly down to a nearly flat potential of 0 in the center region. The motion of the ions in longitudinal direction can rather be considered as motion between hard walls than motion in a harmonic potential.

The trap depth in longitudinal direction isQΦec,0. Longitudinal trap depth However, the electric field of the endcap electrodes has also compo-

nents in thexy direction which have to be considered with respect to the secular potential. The field caused by the endcap electrodes in the center region of the trap is plotted in Fig. 2.8.

We find that the influence of the endcap electrodes is greatly sup- pressed, even though there is a small potential variance among different radial directions (e. g.xandxy). For different trap set-ups (i.

e. those with shorter center electrodes [32]) this influence has to be taken into account in the secular potential (2.27) such as the static voltageU. In our case the endcap potential does not influence the motion in the xy plane of the trap significantly.

Before the dimensions and voltages of the trap will be discussed, in the next section an intrinsic rf trap heating mechanism needs to be mentioned.

2.2.4 Radiofrequency (rf ) Heating

A heating mechanism calledrf heating occurs, when ion–ion or ion–

residual-gas collisions happen in trapping regions of strong rf field.

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F_ec(0, 0, )z

FFecec, 0() /x, y, z

F_ec( , 0, 0)x

F_ec(x x, ,0)

0 1 2 3 4 5 6 7

0,00000 0,00005 0,00010 0,00015 0,00020

distance from trap center mm

F_ec(x, y,0) /F_{ec, 0}

y

x

Figure 2.8: Electric field of the endcap electrodes in the Center of the Trap

The collisions mainly transfer kinetic energy from the micromotion (i. e. energy of the rf field) to the secular motion of the other particle. This brings the ions out of phase with the rf field, the energy cannot be given back to the rf field, but new micromotion energy will be gained from the field. In this sense the ions are heated up by the rf field.

If more than one ion are trapped, the Coloumb interaction will couple the individual ion motions and the space charge effects will change the potentials in the trap. The coupling is strongly dependent on the trap parameters and ion temperature as well as the number of trapped ions. The dynamics of a closely coupled ion plasma in the trap is highly non-linear. Due to the fast rf frequency on the one hand and the long range coulomb interaction on the other hand computer simulations are extremely time consuming and restricted to only a few particles.

In a linear Paul trap the rf field along the z-axis vanishes and so does the micromotion (2.18). Ions which move far off the z-axis gain a considerable kinetic energy due to the micromotion. One advantage of the linear trap design is that the trapping space with low rf field is spread along the z axis instead of a single point in space for the hyperbolic trap design. For this reason the linear Paul trap can store more ions at less kinetic energy. Therefore the mean second order Doppler broadening in a linear trap of length L compared to that in a hyperbolic trap of characteristic size R is reduced by a factor [34]

5R

3L ¿1. (2.30)

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2.3 Experimental Set-up of a Linear Paul Trap 23

Since the rf heating increases with the distance from the trap axis, the heating effect is stronger for particles far off the axis. Particles will be lost if they are heated above the trap depth. Thus the rf heating limits the maximum extension of the ion cloud in transverse direction.

By the rf heating and due to Coulomb coupling of the ions motions the cloud is heated up to a temperature in the order of the trap depth. These energies of typically a few eV correspond to ion temperatures in the order of 10 000 K. A quantitative description of the energy distribution in the trap requires complicated models of interacting particles as described in [35], or a description of the brownian motion in the ion cloud [36, 37, 38], or a quantum motion description as in [39].

With respect to the spectroscopy, the Doppler broadening needs to be reduced in an efficient way by a cooling mechanism. Before the cooling mechanism is discussed in chapter 3 the following sections in this chapter explain the experimental set-up of the trap in detail.

2.3 Experimental Set-up of a Linear Paul Trap

In this section the experimental set-up of the trap, the UHV environment, the trap electronics, the sources for the atoms and molecules, the devices for detection, and lots of auxiliary components are described in detail.

2.3.1 The UHV Environment and Components

The experiment takes place in an UHV environment. The complete set-up consisting of a vacuum chamber, connected pumps, residual gas analyzer, and mounting to the optical table was designed using CAD and is shown in Fig. 2.9.

The vacuum system is designed to achieve extreme ultra-high- vacuum (XUHV) in the 10⁻¹¹ mbar region or below. This is especially important since we trap light particles like HD⁺ or Be⁺, which are typically lost after a single collision. The interaction of the trapped particles with the residual gas then drops to below 1 per 100 s. Disturbances on this time-scale will neither affect the particles lifetime in the trap nor the precision of high resolution spectroscopy.

The chamber was custom made by Vacuum Generators according to our specifications. Detailed drawings of the chamber are given in Fig. 2.10.

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Figure 2.9: CAD designed UHV set-up. The chamber is drawn grey, the pumps and valves blue, the optical table violet, the holder red, elastic parts cyan, the residual gas analyzer pink.

The main body of the chamber consists of a 200 mm diameter, UHV Chamber

486 mm long tube. After fabrication the chamber was heat treated at 850^◦ for 1 hour to reduce the hydrogen concentration in the stainless steel (316LN, 316L, µr < 1.005) and leak checked to be better than 1·10⁻¹⁰ mbar/ls. The inner walls of the chamber are mechanically polished.

The chamber is mounted on a holder in a position that the main operating plane of the chamber is 75 mm above the level of the optical table, which is the general beam height of the whole optical set-up. Sand is used to fill the steel carriers of the mount to damp vibrations. The vacuum set-up of the trap is hung down from the

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Figure 2.10: CAD Drawing of the UHV Chamber according to our specifications by Vacuum Generators. All dimensions in mm.

DN200CF top flange. The electronic feedthroughs connecting the trap electrodes are also inserted in the top flange, so the trap set-up can be taken out in one piece for assembly. It will be described in more detail in (→2.3.2).

The chamber design includes many flanges. In the operating plane optical access is possible every 45^◦. Two orthogonal axes have large DN100CF flanges on short ports providing a large access angle, e.

g. for detection. Several other axes are provided for future experimental set-ups. Components such as ion gauges, the electron gun, leak valves, etc. can be attached to any of the DN40CF flanges.

For ease of use the chamber has tapped flanges and silver-plated screws¹.

Two pumps are connected to the chamber from below the table. Pumps The DN200CF bottom flange is adapted to an ion pump which

has an integrated titanium sublimation pump (typeVarian, VacIon Plus 300, noble diode, the nominal pumping speed of the ion pump is 260 l/s without titanium sublimation). Due to the weight of the pump, the pump sits on suspension elements elements mounted on an adjustable plate.

A small turbo molecular pump (type Varian, V70LP, the nominal

1Technical remark: on the M8 screws, a torque of 25 Nm is applied. On the M6 screws, the torque should be 13 Nm. However, if a tap should ever be damaged beyond, there is clearance to put a nut between the flange and the wall of the chamber. . .

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pumping speed is 70 l/s) is attached to a DN63CF flange via an all- metal right angle valve, in series with a membrane pump . Pumping is started by the turbo- and membrane-pump. When the pressure drops below 10⁻⁵ mbar the ion pump can be started. Due to the finite compression ratio of the turbo-pump the right angle valve needs to be closed when the pressure drops below 10⁻⁸ mbar.

With a clean UHV set-up and if the chamber was carefully vented with nitrogen, the chamber can be pumped down to the low 10⁻¹¹ mK-region within half a day. The UHV system is designed for a bakeout temperature of at least 300^◦C, except for the flange Bakeout

temperatures of the turbo pump that needs to be cooled to stay below a temperature of 120^◦C, and the electronics of the residual gas analyser, which have a maximum temperature of 70^◦C.

For vacuum analysis a pressure gauge (type Varian, UHV-24p us- Vacuum analysis

ing controller of type Granville-Phillips 350) is attached. Due to the small shape of this ion gauge it has a low x-ray limit of 6·10⁻¹² mbar. Furthermore a residual gas analyzer (type Vacuum Generators, Smart IQ+100D) is attached. It is used for more detailed vacuum analysis, leak checking, and to control the bakeout procedure.

An all-metal leak valve (type Vacuum generators, ZVLM263R) is Leak valve

attached for defined gas inlet, e. g. HD. It allows reproducible leak rates down to 1·10⁻¹¹ mbar l/s and can be used as inlet for molecules in gas phase (e. g. H2, HD) as well as for buffer gas inlet of He to cool the ions in the trap. If sympathetic cooling is performed, the molecules are added to a laser-cooled atomic ion cloud in the trap. This process requires low partial pressure and low leak rates in order to avoid destruction of the trapped ion cloud. Further details about cooling in the trap are given in chapter 3. Since recently, a motorized version of the leak valve is available. As soon as loading of the trap becomes computer controlled, the valve should be replaced with the motor driven version.

When peripheral components are (re-)attached to the chamber, it is Peripherals

often necessary to pump away the air inclusions in the connecting volume before opening a valve to main chamber. Therefore two auxiliary valves (type Nupro) are provided on a welding flange, which is inserted between the right angle valve to the chamber and the turbo pump. If the turbo pump is disconnected from the chamber with the right angle valve, it can be used to pump out other volumes via the two auxiliary valves without disassembling.

The window material of the viewports has to be chosen with respect Viewports

of the light to be transmitted. The cooling light at a wavelength of 313 nm requires fused silica (suprasil) windows. If a spectroscopy laser is added (1.4 µm or 2.8 µm) they need to be replaced with sapphire windows. However, for the beginning of the experiment

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and the work presented in this thesis, we have installed less expen- sive suprasil windows (type MDC, AR coating from Ferroperm), which have an additional advantage of no birefringence compared to sapphire. Other window materials (e. g. fluoride) are only barely suitable for use in UHV, because their thermal expansion coefficient is very different compared to the one of the steel housing, and thus restricts the bakeout to low temperatures and very slow temperature changes.

2.3.2 Set-up of the Trap

To reach the goal of high resolution spectroscopy of molecular ions Requirements the linear rf trap should fulfill the following requirements.

• Trap a large number of ions to increase the spectroscopy signal to noise ratio.

• Cool the ions to temperatures down to the 1 Kelvin region or below to reduce the second order Doppler shift sufficiently.

• Allow good optical access to the particles in axial direction for the cooling and spectroscopy laser beams as well as in transverse direction for fluorescence imaging of the laser- cooled ion cloud.

• Provide a quadrupole fieldwith sufficient precision.

• Trap atomic masses 1-9 at a moderate rf voltage and frequency.

• Use XUHV compatible construction and materials for the trap.

The trap designed and built up in this work matches all these requirements and is photographed in Fig. 2.11.

Construction, Dimensions, and Parameters

There is a large number of applications for ion traps and a simi- larly large variety of trap designs [40]. The dimensions of existing quadrupole traps are quite diverse: there are miniature quadrupole traps which trap single or only a few ions which have a very strong trapping potential and a characteristic size of 100 µm [41], there are traps with diameters of several centimeters [26, 42].

A large trapping region is required to trap many ions. Luckily, a large trap also simplifies the optical access into the trap and

Development of an experiment for trapping, cooling, and spectroscopy of molecular hydrogen ions